On Mar 13, 3:09 pm, Zuhair <zaljo...@gmail.com> wrote:> On Mar 12, 10:46 pm, Zuhair <zaljo...@gmail.com> wrote:>>>>>> > On Mar 12, 2:49 pm, Zuhair <zaljo...@gmail.com> wrote:>> > > On Mar 11, 11:17 pm, Zuhair <zaljo...@gmail.com> wrote:>> > > > Let x-inj->y stands for there exist an injection from x to y and there> > > > do not exist a bijection between them; while x<-bij-> means there> > > > exist a bijection between x and y.>> > > > Define: |x|=|y| iff x<-bij->y>> > > > Define: |x| < |y| iff x-inj->y Or Rank(|x|) -inj-> Rank(|y|)>> > > > Define: |x| > |y| iff |y| < |x|>> > > > Define: |x| incomparable to |y| iff ~|x|=|y| & ~|x|<|y| & ~|x|>|y|>> > > > where |x| is defined after Scott's.>> > > > Now those are definitions of what I call "complex size comparisons",> > > > they are MORE discriminatory than the ordinary notions of cardinal> > > > comparisons. Actually it is provable in ZF that for each set x there> > > > exist a *set* of all cardinals that are INCOMPARABLE to |x|. This of> > > > course reduces incomparability between cardinals from being of a> > > > proper class size in some models of ZF to only set sized classes in> > > > ALL models of ZF.>> > > > However the relation is not that natural at all.>> > > > Zuhair>> > > One can also use this relation to define cardinals in ZF.>> > > |x|={y| for all z in TC({y}). z <* x}>> > > Of course <* can be defined as:>> > > x <* y iff [x -inj->y Or> > > Exist x*. x*<-bij->x & for all y*. y*<-bij->y -> rank(x*) in> > > rank(y*)].>> > > Zuhair>> > All the above I'm sure of, but the following I'm not really sure of:>> > Perhaps we can vanquish incomparability altogether>> > If we prove that for all x there exist H(x) defined as the set of all> > sets hereditarily not strictly supernumerous to x. Where strict> > subnumerousity is the converse of relation <* defined above.>> > Then perhpas we can define a new Equinumerousity relation as:>> > x Equinumerous to y iff H(x) bijective to H(y)>> > Also a new subnumerousity relation may be defined as:>> > x Subnumerous* to y iff H(x) injective to H(y)>> Better would be>> x Subnumerous* to y iff H(x) <* H(y)>> however still this won't concur incomparability completely>> However if we define recursively H_n(x) then we can define> the above relations after those. However still incomparability> would persist, although the above is still a strong approach> against incomparability.>>>>>> > This might resolve all incomparability issues (I very highly doubt> > it).>> > Then the Cardinality of a set would be defined as the set of all sets> > Equinumerous to it of the least possible rank.>> > A Scott like definition, yet not Scott's.>> > Zuhair- Hide quoted text ->> - Show quoted text -- Hide quoted text ->> - Show quoted text -

Once again, just like I (and others now) said - you throw out stuffthat isn't ready, you make mistakes, you change it, you ponder how tofix it, you debate possible solutions. It is a total moving target.It is presented as being a solution but it is just a big problem -trying to keep it straight as you move things around in a desperateattempt to clean up the mess.